Chapter 6 · Section 5 of 8
Find the temperature dependence for the Hubble parameter in the early flat Universe.
In early Universe the total energy density is determined by by the relativistic particles. Therefore $\left( \hbar = c = 1\right)$: \[ H^2 = \frac{8\pi G}{3}\rho;\ \rho = \frac{\pi ^2}{30}g^* T^4;\ G = \frac{1}{M_{Pl}^2}\Rightarrow H =(2\pi)^{3/2}\sqrt{\frac{g^*}{90}}\frac{T^2}{M_{Pl}}\approx 1.66\sqrt {g^* } \frac{T^2}{M_{Pl}}.\]
Find the time dependence of the temperature of the early Universe by direct integration of the first Friedman equation.}
\[\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho ;\quad \rho = \alpha T^4 ;\quad aT = const = A\Rightarrow a\dot a = A^2 \left(\frac{8\pi G\alpha}{3}\right)^{1/2}.\] Integrating the latter equation with the initial condition $a(t = 0) = 0$ one obtains \[a(t)=A\left(\frac{32\pi G\alpha}{3}\right)^{1/4}\sqrt{t}\Rightarrow T = \left( \frac{3}{32\pi G\alpha} \right)^{1/4} t^{-1/2}.\]
Use the relation \[ \alpha = \frac{\pi ^2}{30}g^*, \] which follows from the comparison of the equations $\rho = \alpha T^4 $ and \[\rho = \frac{\pi^2}{30}g^* T^4.\]
Determine the energy density of the Universe at the Planck time.
\[ \rho = \frac{3M_{Pl}^2}{8\pi}H^2 ;\quad H = \frac{1}{2t}; M_{Pl} = \frac{1}{t_{Pl}} \Rightarrow \rho \left( t = t_{Pl} \right) = \frac{3M_{Pl}^4}{32\pi} \approx 6 \cdot 10^{74} \,GeV^4.\]
Show that at Planck time the energy density of the Universe corresponded to $10^{77}$ proton masses in one proton volume.
Taking into account that $0.2\,GeV \cdot \, fm = 1$ and using the result o previous problem one obtains \( \rho (t = t_{Pl} ) \approx 6 \cdot 10^{74} \, GeV^4 \approx 0.75 \cdot 10^{77} \,GeV/\,fm^3, \) which approximately corresponds to $10^{77}$ proton mass inside the volume of a proton.
What was the temperature of radiation-dominated Universe at the Planck time?
Using the result of the previous problem one obtains \[T(t=t_{Pl})=\left(\frac{\rho(t=t_{Pl})}{\alpha}\right)^{1/4}\simeq 6.3\cdot10^{31}\, K.\]
Determine the age of the Universe when its temperature was equal to $1\ MeV$.
In problem of the present Chapter it was obtained that \[ t \simeq \frac{0.301}{\sqrt {g^*}}\frac{M_{Pl}}{T^2}.\] At temperature $1\, MeV$ the relativistic particles are presented by photons, electrons, neutrinos of all three types and their antiparticles. Therefore \[ g^* = 2 + \frac{7}{8}(4 + 2 \times 3) = 10.75. \] Taking into account that $1/\,GeV \approx 0.7 \times 10^{- 24} \,s $ one obtains \[ t(T \approx 1\,MeV) \approx 0.75\, s.\]
In the first cyclic accelerator - the cyclotron (1931)- particles were accelerated up to energies of order $1MeV$. In the next generation accelerators - the bevatrons - energy was risen to $1GeV$. In the last generation accelerator - the LHC -protons are accelerated to energy of $1\ TeV$. What times in the Universe history do those energies allow to investigate?
Show that in the epoch when the energy density of the Universe was determined by ultra-relativistic matter and effective number of internal degrees of freedom did not change, held $\dot{T}/T\propto -T^2$.
For the ultra-relativistic component \[ H \propto T^2 ,\quad T \propto a^{ - 1} ,\quad \dot T \propto - \frac{\dot a}{a^2}\Rightarrow \frac{\dot T}{T} \propto - H \propto - T^2.\]
Estimate the baryon-antibaryon asymmetry $A\equiv(n_b-n_{\bar{b}})/n_{\bar{b}}$ in the early Universe.
The early Universe means the period of evolution of Universe, after which there are no processes capable to violate the baryon number conservation law. Than the total baryon number in a comoving volume will be constant. Therefore $$ \left(n_b-n_{\bar{b}}\right)a^3=\left(n_{b0}-n_{\bar{b}0}\right)a_0^3. $$ However today the anti-baryons are practically absent, thus $n_{\bar{b}0} \approx 0,$ and therefore $$ A \equiv \frac{n_b-n_{\bar{b}}}{n_{\bar{b}}} = \frac{n_{\bar{b}0}}{n_b}\frac{a_0^3} {a^3}. $$ Using the estimate $a \sim 1/T,$ represent the expression for asymmetry in the following form $$ A \approx \frac{n_{\bar{b}0}}{n_b}\frac{T^3} {T_0^3}. $$ The densities of photons and baryons are connected to temperature by the relations ${n_b} \approx {T^3}$ and $ n_{\gamma0} \approx T_0^3 $ (the numerical factors of order of unity are omitted). Using the above given ingredients, the asymmetry can be presented in the following form $$ A \approx \frac{n_{b0}} {n_{\gamma0}}. $$ Therefore the baryon-anti-baryon asymmetry equals to current ratio of the baryon number density to that of photon number. More rigorous analysis leads to the relation \[A \approx 6\frac{n_{b0}}{n_{\gamma0}}.\] The current photon number density is well defined by the CMB temperature and equals to $410.4\: cm^{ - 3}.$ The baryon number density can be estimated by several ways, for example, basing on relative abundance of hydrogen and deuterium. The ultimate result reads $$ A \approx 3 \cdot 10^{ - 9}. $$ The latter result can be interpreted in the following way: in early Universe there were three extra quarks per each billion of anti-quarks. presently observed matter is nothing that the result of that tiny asymmetry.
Determine the monopoles' number density and their contribution to the energy density of the Universe at the great Unification temperature. Compare the latter with the photons' energy density at the same temperature.
Consider two regions situated so far from each other that they are not causally connected. Thus the regions take generally speaking independent configurations. Therefore a so-called topological defect, analogous to a dislocation in ferromagnetic crystals, appears on the boundary between the regions. A simplest type of such a defect is analogous to a point dislocation. In typical Grand Unification Theories (GUT) such objet represents the magnetic monopole. It behaves as a particle with mass $$ m_{mon}\approx \frac{M_X}{\alpha_U}. $$ Here ${M_X}\approx 10^{16}\,GeV$ is the GUT energy scale and $\alpha_U\approx 1/40$ is the effective coupling constant. The magnetic monopoles could be created in the hot Universe at the phase transition connected to spontaneous symmetry breaking: when temperature of the Universe falls lower than $T_c\approx E_{GUT}\approx 10^{16}\,GeV,$ the Higgs field presented by $X$ and $Y$ massive bosons acquires non-zero vacuum mean. Due to their giant mass, the monopoles become nonrelativistic component of the energy density in the Universe right after their creation. The monopole density is expected to roughly equal to unity in each isolated region. Size of such region is determined by the distance passed by light during the time period $t_c$ from the Big Bang to the phase transition. This distance simply equals to the particle horizon at time $t_c.$ If the universe was dominated by radiation up to that time, then $a\sim t^{1/2}$ and therefore the particle horizon equals to $a(t)\int_0^{t_c}\frac{dt'}{a(t')}=2t_c$. Then the predicted monopole number density reads $$ n_{mon}\approx \frac{1}{\left( 2t_c \right)^3} $$ The time period ${t_c}$ can be estimated from the relation $$ t_c=\frac 14\sqrt{\frac{45}{\pi^3g^*}} M_{Pl}T_{c}^{-2}\approx 10^{-39}\, sec. $$ As the monopoles are non-relativistic particles, then their contribution into the energy density is $$ \rho_{mon}=n_{mon}m_{mon}\approx \frac{1}{\left( 2t_c \right)^3}\frac{M_X}{\alpha_U} \approx 2\times 10^{57}\,GeV^4. $$ Compare this value to the photon concentration at the same time $$ \rho_{\gamma }=\frac{\pi^2}{15}T_c^4\approx 2\times 10^{63}\,GeV^4. $$
At what temperature and time does the contribution of monopoles into the Universe energy density become comparable to the contribution of photons?
As it was shown in the previous problem, initially the photon energy density considerably exceeded that of monopoles: $\rho_{\gamma }/\rho _{mon}\approx 10^6$. However as the photons are relativistic particles then $\rho _\gamma\sim 1/a^4,$ while the monopoles are non-relativistic and $\rho_{mon}\sim 1/a^3.$ The two energy densities become equal when the scale factor increases in $10^6$ times. Correspondingly the temperature decreases in $10^6$ times, as $a\sim 1/T.$ Using the relation $t\sim T^{-2},$ one obtains that the equality between $\rho_\gamma$ and $\rho_{mon}$ takes places after time increases by factor $10^{12}$. Thus, starting from the GUT scale, $T\approx 10^{16}\,GeV$ at times of order of $t\approx 10^{-39}\,sec,$ the equality $\rho_{\gamma }=\rho_{mon}$ is expected to take place at temperature $T\approx 10^{10}\,GeV$ and time $t\approx 10^{-27}\,sec.$